p-adic Analysis

The research unit PEAD studies Non-archimedean functional analysis. The term "non-archimedean" refers to the fact that the scalars, which in classical functional analysis are real or complex numbers, are now elements of a valued field K with non-archimedean valuation. I.e. a valuation that satisfies the strong triangle inequality |x+y| £ Max{|x|, |y|}. The best known examples of such fields are the fields of p-adic numbers, p a prime number (The theory is therefore known under the name "p-adic functional analysis"). In such a setting one can develop a theory of Banach spaces, locally convex spaces, lineair operators etc..., just as in the classical case. This non-archimedean theory turns out to be completely different from the classical functional analysis. Not only as far as the results are concerned but also with respect to the techniques needed for the proofs. Also the results depend on the structure of the field of scalars. At this moment, research is concentrated on the study of the extension of p-adic compact operators, inductive limits, Schauder bases, spaces of continuous functions, p-adic distributions and the Fourier Transform. This research is part of an international project. (Belgium, Spain, Poland, Russia and the Netherlands).